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The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
An Efficient Routing Algorithm for Mesh-Hypercube
(M-H) Networks
Abstract - This paper presents an efficient routing algorithm
for the Mesh-Hypercube (M-H) network. The M-H network is
one of the new interconnection networking techniques use to
build high performance parallel computers. The combination
of M-H networks offers high connectivity among multiple
nodes, fault-tolerance, and load scalability. However, the
performance of M-H networks may degrade significantly in
the presence of frequent link or node failures. When a link or
node failure occurs, neither the hardware schemes nor point
to point and multistage routing algorithms can be used
without adding extra links. This paper presents an efficient
single bit store and forward (SBSF) routing algorithm for M-
H network that based on the round robin scheduling
algorithm. Simulation and numerical results suggest that the
proposed routing algorithm improves the overall performance
of M-H network by both reducing the transmission delay and
increasing the total data throughput even in the presence of
faulty nodes.
Keywords: High performance systems, Mesh-Hypercube
networks, routing algorithm, fault-tolerance, and load
scalability
1 Introduction
A key component of high performance parallel computer
architecture is the interconnection network topology which
enables the communication among different nodes of a large
parallel computer. Several different interconnection network
topologies have been proposed over the years [1, 2, 3]. One of
the most popular, new, and an efficient network topology is
the mesh-hypercube (M-H) networks. This network has been
used in different systems such as n-Cube, Intel IPSC [13, 14,
15].
With the continuous growth in network size, it is hard to
find a large size network such as an M-H network without
the presence of faulty nodes and or links. It has been shown
that the performance of M-H networks degrades as the
number of faulty links increases in a large parallel network
[11, 12]. When the numbers of faulty links or nodes increase
in the parallel system, the non-faulty reachable nodes of the
network decrease proportionally. This, therefore, results
performance degradation due to the lack of efficient routing
algorithm that can provide full reachability to each non-faulty
node of the parallel network.
This paper presents a new and an efficient single bit
store and forward (SBSF) routing algorithm for M-H network
that based on the round robin scheduling algorithm. The
proposed SBSF routing algorithm can be implemented on a
large parallel network which may consist of multiple nodes
where each node can send and receive random point-to-point
messages. The primary objective of this paper is to simulate
the data throughput of an M-H network as well as its
performance in the presence of faulty nodes and or links and
hot spots. In addition, the simulation results of this paper
allow one to examine the impact of different architectures on
the overall performance of the M-H networks. These different
architectures include dimension of the M-H networks, input
parameter values such as the variation in the total message
generation rate per clock cycle, and the introduction of faulty
links. We show that the performance of M-H networks
remains consistent even in the presence of multiple faulty
nodes. Our numerical analysis shows that the proposed
algorithm maintains a consistent performance even in the
presence of faulty nodes by making non-faulty nodes fully
reachable.
The combination of M-H network offers several
attractive features. First, it has high connectivity between the
various processors [5]. In a P processor system, a message
must traverse no more than log2 (P) links before reaching its
final destination. Second, it is a multiple instructions
multiple data (MIMD) machine, allowing different nodes of a
network to implement different component models [2]. Third,
the architecture has good scaling properties for a large value
of P [3].
As the numbers of nodes grow in an M-H network, the
number of links required at each node increases
logarithmically where as the required communication
bandwidth increases linearly. Nodes in an M-H network can
also be viewed as a directed graph where all interconnections
between the nodes occur in a synchronous order. In addition,
for the sake of experimental verification, we assume that each
link in an M-H network may carry no more than one unit
Syed S. Rizvi
1
, Khaled M. Elleithy
1
, and Aasia Riasat
2
1Computer Science and Engineering Department, University of Bridgeport, Bridgeport, CT USA
2Department of Computer Science, Institute of Business Management, Karachi, Pakistan
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
message in one step where each node during a step can send
at most one message to each of its neighboring node.
The remainder of this paper is organized as follows.
Section 2 presents some of the well known routing
algorithms for M-H networks. Section 3 presents the
proposed SBSF routing algorithm along with a
comprehensive discussion on its implementation. In Section
4, we examine and simulate the data throughput and the hot
spot properties to determine the overall performance of the
proposed algorithm. Finally, section 5 concludes the paper.
2 Routing algorithms for M-H networks
Substantial efforts have been devoted to the study of
message routing for M-H networks [1, 4, 5]. Routing
algorithms for M-H network are broadly classified into two
categories: oblivious routing and adaptive routing [9, 10]. In
an oblivious routing algorithm, the path of one message is
unaffected by the presence of other messages in the network
[6]. On the other hand, in an adaptive routing algorithm,
messages may be directed away from the congested parts of
the network [5, 7]. Optimal routing for given paths on
arbitrary networks has been studied extensively in the context
of store-and-forward algorithms. Routing algorithms, in
which packets do not strictly follow specific paths, are
matching routing and hot-potato routing [7, 8].
In randomized routing, a message is sent from a source
to a destination in two stages [9, 10]. In the first stage, the
message is sent from the source to a random intermediate
node. In the second stage, the intermediate node forwards the
received message to the intended destination. In each stage of
randomized routing, the message is routed using the bit-
fixing algorithm, which is also known as dimension-order
and the e-cube routing algorithms. The randomized routing
algorithms are fast and can solve any one-to-one packet
routing problems on a P-node hypercube in the order of O
(log2 P) packet steps [9]. One drawback to such algorithm,
however, is that they all require in order of O (log2 P) bit
steps when implemented on real parallel machines such as
the connection machines. Since packets always contain at
least in order of log (P) bits of addressing information, a
typical packet step really consists of O (log P) bit steps [10].
In bit-fixing routing algorithm, dimensions are ordered
in an arbitrary way and a message is always typically directed
along the lowest dimension of the network in which its
current position and its final destination differ [11]. A simple
implementation of this algorithm is shown in Fig. 1. The
routing via bit-fixing algorithm can take exponential time.
Consider a permutation on the n-cube where n = 4 and to
route a packet from 0000 to 1111, the bit-fixing routing
algorithm uses the following path:
0000 1000 1100 1110 1111
→→→→. It has been shown [9] that
this simple permutation causes the bit-fixing routing
algorithm to take in order of
2
(2 )
n
Ωsteps to route a
message. From the performance point of view, the resultant
computational complexity is not desirable.
3 Proposed SBSF routing algorithm for
the M-H networks
We propose an efficient SBSF message routing
algorithm for the M-H networks which based on a simple
round robin scheduling algorithm. The proposed SBSF
algorithm shows some good load-scalability characteristics
among the M-H nodes. One of the reasons for achieving high
scalability is due to the use of round robin scheduling
algorithm. Instead of separately using a scheduling algorithm
on each output queue, we implement the round robin
algorithm as a part of our proposed routing algorithm. By
doing this, we greatly reduce the average waiting time of
each output queue associated with each outgoing link. Our
choice for using the round robin scheduling algorithm with
the proposed SBSF algorithm is due to the fact that it is both
simple and easy to implement as well as starvation-free. The
advantages of the proposed SBSF routing algorithm such as
the reduced average waiting time per output queue and
consequently the improved data throughput can be seen in
our simulation results.
3.1 Proposed M-H model and assumptions
Before going to present the proposed SBSF routing
algorithm for an M-H network, it is worth mentioning some
of our key assumptions:
1. We assume that the total number of nodes ‘P’ in the
M-H network is equal to 2k where k represents the
dimension of a hypercube.
2. The total nodes P are identified with a unique node
number in the range of 0 to 2k – 1.
3. Every node in an M-H network has k bi-directional
links with the other nodes.
S(j)
0, 0, 0, 0
S(j)
0, 0, 0, 1
S(j)
0, 0, 1, 1
S(j)
0, 0, 0, 0
S(j)
1, 0, 0, 1
Fig.1. Bit-fixing routing algorithm
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
4. Two nodes are said to be connected if and only if their
binary addresses representation differs in exactly one
bit order.
5. The binary representation of a node in an M-H
network with k dimension has k bits (the exception is
the singular case of hypercube with k = 0).
A shortest path between any two nodes of an M-H
network is represented as a sequence of nodes 0 1
, ,......,
i
P P P
where subscript i indicates the length of the path. In an M-H
network, multiple shortest paths may exist between any two
nodes. For instance, if there are P total nodes exist in an M-H
network which is numbered from 0 to P-1 in such a way that
addresses of connected nodes differ exactly by one bit in their
binary representation (see Fig. 2), then the M-H network
exhibits some desirable properties.
It can be seen in Fig. 2 that the routing in the cube is
simplified to find any direct neighbor that reduces the
number of bit differences between the address of the recipient
and the messenger. As an illustration, let P0 and P7 be the
nodes in a k-dimensional M-H network. The total number of
bit positions at which these binary equivalents of P0 and P7
differ can be considered as the hamming distance H. In other
words, the length of the shortest path between P0 and P7 is
equal to the hamming distance H. For instance, in Fig. 2, in
order to route a message from a node P0 (000)2 to a node P7
(111)2, nodes P4 (100)2 and P6 (110)2 provides an optimal
route since at each given node along the path, the bit
differences with P7 (111)2 are reduced by one.
3.2 Proposed SBSF message passing algorithm
Fig. 3 provides the formal description of the proposed
SBSF algorithm. The proposed SBSF routing algorithm starts
by searching nodes in an M-H network. The searching is
performed in a sequential manner starting from the minimum
value of the node-address to a maximum value. For instance,
in a three dimensional hypercube within an M-H network,
the proposed routing algorithm starts searching the node
addresses from 000 and proceeds in a sequential manner until
it reaches the maximum value (i.e., 111). In other words, this
sequential searching of nodes will continue until the node
array becomes empty. Once it reaches to the end of the array,
the algorithm starts over again and this cycle of searching
goes on.
We assume that each node in an M-H network can either
generate or receive a message but not both during the same
clock cycle. Fig. 4 shows the internal architecture of each
node in an M-H network that includes one processor P, one
communication interface C, and four incoming and outgoing
links. Each link in the node architecture has an output queue
for outgoing messages. Also, it should be noted that there is
no input queue for the incoming messages as shown in Fig. 4.
This assumption implies that each incoming message for a
destination node simply goes into sink. We assume that a
node can receive or send a message during a cycle. This
implies that while C receives a message from a different node
through one of its incoming links, it can not send a message
using an output queue from a link different than the one
currently receiving a message as shown in Fig. 4. This
assumption further leads us to the following two facts: An
incoming message for a destination node can either directly
go into an output queue or be absorbed by the processor. A
message needs to wait at least one cycle before it can be sent
once it is received. In other words, you cannot receive a
message and send it out during the same cycle as if the node
did not exist.
For the sake of simulation, we assume that an M-H
network can have at most 256 nodes at one time. In other
words, for ease of understanding, our simulation supports at
most 8 dimensional hypercube within an M-H network that
can have at most 256 nodes. In addition, step 2.3 of Fig. 3
shows the use of round robin scheduling algorithm for
selecting appropriate communication link to receive
messages. For instance, when a certain node is in the busy
state (i.e., the current state of the node is either sending or
receiving or idle), it can not receive messages from other
nodes. Once the proposed routing algorithm gets a free-node,
it implements the round robin scheduling algorithm to check
the status (Busy or Free) of each outgoing link attached to the
output queue. These communication links are further
attached to other nodes as shown in Fig. 5.
Once the proposed SBSF algorithm gets a non-busy link,
it performs the following tasks. First, it checks the output
queue of the attached node (destination node) to determine is
there any message left in the queue. If the output queue does
not contain a single message, the round robin scheduling
P
0
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
0
= 0 = 000
P
1
= 1 = 001
P
2
= 2 = 010
P
3
= 3 = 011
P
4
= 4 = 100
P
5
= 5 = 101
P
6
= 6 = 110
P
7
= 7 = 111
For example:
P
0
= 0 = 000 can go to P
1,
P
2,
and
P
4
. This can also be represented
as:
P
0
=0=000 P
1
= 1 = 001
P
2
= 2 = 010
P
4
= 4 = 100
H. Distance
One Bit
Fig.2. Numbering scheme for three dimensional M-H
networks
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
algorithm will move to the next immediate communication
link. Secondly, if the output queue contains at least one or
more messages, the sinking node starts retrieving the
message(s) from the output queue of the attached destination
node. Finally, the round robin scheduling algorithm changes
the status of both the sinking node and the communication
link and set them as busy.
The important point that we should note here is the
advancement of the round robin scheduling algorithm from
the current link to the next communication link. If we do not
advance to the next link, the proposed SBSF routing
algorithm does not achieve the load-scalability that results
higher values of average waiting time in each output queue.
The higher values of average waiting time in each output
queue would likely to reduce the overall data throughput of
an M-H network.
4 Experimental verifications and the
simulation results
We categorize our performance analysis of proposed
SBSF routing algorithm for an M-H network in two parts. In
Formal Specification of Proposed SBSF Routing Algorithm for M-H Networks
1- Starts searching the nodes
Search Node_ Address [Minimum] to Node_ Address [Maximum]
2- Check the status of each visiting node
2.1. [Set Node status]
Set Node Status = = Sending or
Set Node Status = = Receiving
[Repeat Step 2.2 or Step 2.3] While Node ≠ NULL
2.2. [Check the Condition] If Node Status = Sending Then
Set Node Status = = Creating a Message or
Set Node Status = = Receiving a Message or
Set Node Status = = Sending a Message or
Set Node Status = = Idle
[End of Step 2.2]
[Go back to step-2] Advanced to Next Node
2.3. [Check the Condition] If Status = Receiving Then
2.3.1. Implement Round Robin Scheduling Algorithm
[Start Sequential Search for Node Status]
2.3.1.1. If Node = Busy Then
2.3.1.1.1. Advanced to Next Node
2.3.1.1.2. Go back to Step 2.3.1
2.3.1.2 If Node ≠ Busy Then
[Start Sequential Search for each outgoing link status]
2.3.1.2.1. If Link = Busy Then
Advanced to Next Lin
2.3.1.2.2. If Link ≠ Busy Then
[Get message from the free-node]
[Change the status of each node]
Set Sink Node ==BUSY
Set Destination Node = BUSY
2.3.1.2.3. Advanced the Link
2.3.1.2.4. Go back to Step 2.3.1
[End of Step 2.3(Round Robin Scheduling Algorithm for Message Reception)]
[Go back to Step 2] Advance to Next Node
2.4. [End of Step-2]
[Go back to Step-1]
3- [End of Step-1]
[EXIT]
Fig.
3
.
Formal description of the proposed SBSF routing algorithm
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
the first part, we present our performance analysis when all
communication links are in the operational mode where as in
the second part faulty links or nodes will be considered.
4.1 Performance analysis in the absence of
faulty links or nodes
For this section, a three dimensional M-H network is
considered that can have at most 8 nodes. Furthermore, we
run all simulations for 10000 cycles. In addition, the hot spot
property is also set to zero (i.e., the probability of creating hot
spot is set to zero to ensure an ideal condition for an M-H
network).
Fig. 6 shows the average waiting time in the output
queue and the maximum output queue size with respect to a
range of message transmission probabilities. Average waiting
time in the output queue is the amount of time by which each
packet needs to wait until it reaches to the final destination
node as shown in Fig. 6. On the other hand, the maximum
link output queue size is the size found on any given node
during the simulation. As we increase the probability of
message transmission, nodes in M-H network create more
messages and send them to one or more output queues where
they reside until they get a chance to transmit to other nodes.
It can be seen in Fig. 6 that the increase in probability of
message transmission results a large number of message
creation which might make the output queue congested. As
the output queue becomes congested due to a linear increase
in the probability of message transmission, the average
waiting time per node in the output queue increases
significantly. In addition, the simulation result of Fig. 6
proves the correctness of our proposed SBSF routing
algorithm.
Fig. 7 shows the data throughput with respect to the
probability of message transmission. The data throughput is
the product of utilization per node and the total messages
received successfully. In harmony to our expectations, the
reduction in the messages received per node causes a
reduction of data throughput for the M-H network. It should
be noted that the data throughput in Fig.7 represents the
number of messages received per node in total cycles (i.e., we
set the design parameter CTotal 10000 cycles for all
simulations that we run). For instance, according to our
numerical analysis, when we have 25% of probability, the
total number of messages received is around 13043 that
generate approximately 4891 messages per node. This
implies that each node processes approximately half
messages (0.489 messages) of information per clock cycle. It
can be seen in Fig. 7 that the throughput reduces due to the
reduction in the total messages received successfully.
4.2 Performance analysis in the presence of
faulty links or nodes
For this section, we introduce faulty links and nodes. As
one can easily observe that the introduction of faulty links
and nodes in M-H network reduces message transmission and
Fig.4. Internal architecture of a node in a hypercube
network
Fig.5. Implementation of the proposed SBSF message
passing routing algorithm
0 10 20 30 40 50 60 70 80 90 1 00
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Average Waiting Time (Cycles) and th e Maxim um Queue S ize (Me ssages ) vers us
Probability of M essage Transm issio n
Probablity Of Tr ansmis sion (%)
A ve ra ge W a iti ng T i m e & T he S iz e O f O u t pu t Q u e ue
Avg Waiting time
Max Queue Size
Fig.6. Average waiting time (Cycles) and the maximum
queue size (Messages) versus probability of message
transmission
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
reception as well as a slight decrease in overall data
throughput.
By carefully looking at the simulation results of Fig. 8,
one can make a conclusion that broken links in M-H network
are more affected to message reception as compared to
message transmission. With harmony to our expectations, the
simulation results of Fig. 8 shows a nice reduction in both
message transmission and reception.
Fig. 9 shows the simulation results for utilization per
node and the data throughput in the presence of faulty links.
It should be noted in Fig. 9 that an average utilization per
node in the presence of faulty link is the same as in the
presence of no faulty link except at one place. This is due to
the fact that the utilization per node heavily relies on the total
number of messages generated and received. As we have
mentioned in Fig. 8 that the presence of faulty links and
nodes does not have any sever effects on the message creation
as compared to the message reception. This implies that the
message creation in the presence of faulty link dominates the
total number of messages created and received per node and
thus cancels out the effects of reduced message reception on
the utilization.
This is one of the reasons that why utilization per node
remains the same for both faulty and non-faulty link. On the
other hand, the data throughput in the presence of faulty link
has decreased slightly as shown in Fig. 9. It should be noted
that the data throughput is a product of utilization per node
and the total number of messages received successfully. Since
the utilization is same for both faulty and non-faulty links,
the reduction in the total number of received messages due to
a faulty link causes a slight decrease in the data throughput.
However, the overall data throughput performance is almost
overlapping for most of the values of transmission probability
as shown in Fig. 9.
5 Conclusion
In this paper, we presented a new SBSF routing
algorithm for an efficient transmission and reception of
messages between the nodes within an M-H network. Our
simulation results have shown that the proposed algorithm
maintains a consistent performance even in the presence of
faulty nodes or links. In addition, our experimental
verifications suggest that the proposed routing algorithm
improves the overall performance of the M-H networks by
providing scalable performance for large networks. In
addition, both numerical and simulation results presented in
this paper show the effectiveness of the proposed SBSF
0 10 20 30 40 50 60 70 80 90 100
0
2000
4000
6000
8000
10000
12000
14000
Data Throughpu t versu s
Probability of M essa ge Trans missi on
Probablity Of Tran smiss ion (%)
M es sa g es R ec ei ve d & D at a T hr ou g h p ut
Message Receive
Data Throughput
Fig.7. Data throughput versus probability of message
transmission
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
8x 10
4
Mes sage Cr eation and Rece ption Ver sus
Probability Of Me ssage Transm ission With Faulty Lin k
Probablity Of Tran smis sion (%)
M es sa ge s Cr ea te & R ec eiv e Wi th F L & N o FL
Message Create(No FL)
Message Receive(No FL)
Message Create(FL)
Message Receive(FL)
Fig.8. Message creation and reception versus probability
of message transmission with faulty link (FL)
0 10 20 30 40 50 60 70 80 90 100
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Utilization Per Node & Data Through put Ver sus
Probability Of Me ssage Transm ission With Faulty Link
Probablity Of Tr ansmis sion (%)
U til iz at io n Pe r N o de & D at a Th ro u gh p u t Wit h F L & N o F L
Utilization/Node (No FL)
Data Through put(No FL)
Utilization/Node (FL)
Data Through put(FL)
Fig.9. Utilization per node and data throughput versus
probability of message transmission with faulty link (FL)
The 2008 World Congress in Computer Science, Computer Engineering, and Applied Computing. pp. 69 - 75, July 14 - 17, 2008
algorithm by minimizing the average waiting time per output
queue as well as increasing the total data throughput for both
faulty and non-faulty links.
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